The sine and cosine functions

Respond to each of the following prompts. Where a derivative is requested, be sure to label the derivative function with its name using proper notation.

1. Determine the derivative of $h(t) = 3\cos(t) - 4\sin(t)$.

2. Find the exact slope of the tangent line to $y = f(x) = 2x + \frac{\sin(x)}{2}$ at the point where $x = \frac{\pi}{6}$.

3. Find the equation of the tangent line to $y = g(x) = x^2 + 2\cos(x)$ at the point where $x = \frac{\pi}{2}$.

4. Determine the derivative of $p(z) = z^4 + 4^z + 4\cos(z) - \sin(\frac{\pi}{2})$.

5. The function $P(t) = 24 + 8\sin(t)$ represents a population of a particular ki…

Let $f(x) = 3\cos(x) - 2\sin(x) + 6$.

1. Determine the exact slope of the tangent line to $y = f(x)$ at the point where $a = \frac{\pi}{4}$.

2. Determine the tangent line approximation to $y = f(x)$ at the point where $a = \pi$.

3. At the point where $a = \frac{\pi}{2}$, is $f$ increasing, decreasing, or neither?

4. At the point where $a = \frac{3\pi}{2}$, does the tangent line to $y = f(x)$ lie above the curve, below the curve, or neither? How can you answer this question without even graphing the function or the tangent line?

In this exercise, we explore how the limit definition of the derivative more formally shows that $\frac{d}{dx}[\sin(x)] = \cos(x)$. Letting $f(x) = \sin(x)$, note that the limit definition of the derivative tells us that .

1. Recall the trigonometric identity for the sine of a sum of angles $\alpha$ and $\beta$: $\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)$. Use this identity and some algebra to show that .

2. Next, note that as $h$ changes, $x$ remains constant. Explain why it therefore makes sense to say that…